A FUNCTIONAL APPROACH TO d-ALGEBRAS
A FUNCTIONAL APPROACH TO d-ALGEBRAS
The Pure and Applied Mathematics. 2015. May, 22(2): 179-184
• Received : April 17, 2015
• Accepted : May 08, 2015
• Published : May 31, 2015 PDF e-PUB PubReader PPT Export by style
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KEUM SOOK, SO

Abstract
In this paper we discuss a functional approach to obtain a lattice-like structure in d -algebras, and obtain an exact analog of De Morgan law and some other properties.
Keywords
1. INTRODUCTION
Y. Imai and K. Iséki introduced two classes of abstract algebras: BCK-algebras and BCI -algebras ( [8 , 9] ). BCK -algebras have some connections with other areas: D. Mundici  proved that MV -algebras are categorically equivalent to bounded commutative BCK -algebras, and J. Meng  proved that implicative commutative semigroups are equivalent to a class of BCK -algebras. It is well known that bounded commutative BCK -algebras, D -posets and MV -algebras are logically equivalent each other (see [4, p. 420]). We refer useful textbooks for BCK / BCI -algebra to [4 , 6 , 7 , 12 , 17] . J. Neggers and H. S. Kim (  ) introduced the notion of d-algebras which is a useful generalization of BCK -algebras, and then investigated several relations between d -algebras and BCK -algebras as well as several other relations between d -algebras and oriented digraphs. J. S. Han et al. (  ) defined a variety of special d -algebras, such as strong d -algebras, (weakly) selective d -algebras and others. The main assertion is that the squared algebra ( X ; □, 0) of a d -algebra is a d -algebra if and only if the root ( X ; ∗, 0) of the squared algebra ( X ; □, 0) is a strong d -algebra. Recently, the present author with H. S. Kim and J. Neggers (  ) explored properties of the set of d -units of a d -algebra. It was noted that many d -algebras are weakly associative, and the existence of non-weakly associative d / BCK -algebras was demonstrated. Moreover, they discussed the notions of a d -integral domain and a left-injectivity in d / BCK -algebras. We refer to [1 , 2 , 15 , 16] for more information on d -algebras.
In this paper we discuss a functional approach to obtain a lattice-like structure in d -algebras, and obtain an exact analog of De Morgan law and some other properties.
2. PRELIMINARIES
An ( ordinary ) d - algebra (  ) is a non-empty set X with a constant 0 and a binary operation “ ∗ ” satisfying the following axioms:
• (D1)x∗x= 0,
• (D2) 0 ∗x= 0,
• (D3)x∗y= 0 andy∗x= 0 implyx=yfor allx,y∈X.
A BCK -algebra is a d -algebra X satisfying the following additional axioms:
• (D4) (x∗y) ∗ (x∗z)) ∗ (z∗y) = 0,
• (D5) (x∗ (x∗y)) ∗y= 0 for allx,y,z∈X.
Example 2.1 (  ). Consider the real numbers R , and suppose that ( R ; ∗, e ) has the multiplication
x y = ( x y )( x e ) + e
Then x x = e ; e x = e ; x y = y x = e yields ( x y )( x e ) = 0, ( y x )( y e ) = e and x = y or x = e = y , i.e., x = y , i.e., ( R ; ∗, e ) is a d -algebra.
3. A FUNCTIONAL APPROACH TOd-ALGEBRAS
Let ( X , ∗, 0) be a d -algebra. A map ϕ : X X is said to be order reversing if x y = 0 then ϕ ( y ) ∗ ϕ ( x ) = 0 for all x , y X ; self-inverse if ϕ ( ϕ ( x )) = x for all x X ; an anti-homomorphism if ϕ ( x y ) = ϕ ( y ) ∗ ϕ ( x ) = 0 for all x , y X ; a homomorphims if ϕ ( x y ) = ϕ ( x ) ∗ ϕ ( y ) for all x , y X .
Example 3.1. Consider X := {0, a , 1} with PPT Slide
Lager Image
Then ( X ; ∗, 0) is a d -algebra. If we define a map ϕ : X X by ϕ (0) = 1, ϕ ( a ) = a and ϕ (1) = 0, then it is easy to see that ϕ is both self-inverse and order reversing, but it is not an anti-homomorphism, since ϕ ( a ∗1) = ϕ (0) = 1 and ϕ (1)∗ ϕ ( a ) = 0∗ a = 0.
Moreover, it is not a homomorphism, since ϕ (0 ∗ a ) = ϕ (0) = 1 ≠ a = 1 ∗ a = ϕ (0) ∗ ϕ ( a ).
Proposition 3.2. Let ( X , ∗, 0) be a d-algebra . If ϕ : X X is a (anti-) homomorphism, then ϕ (0) = 0.
Proof . Since X is a d -algebra, by ( D 1), we obtain ϕ (0) = ϕ ( x x ) = ϕ ( x ) ∗ ϕ ( x ) = 0.
Proposition 3.3. If ( X , ∗, 0) i s a d-algebra, then every anti- homomorphism is order reversing .
Proof . Let ϕ : X X be an anti-homomorphism. If we assume that x y = 0, then ϕ ( y ) ∗ ϕ ( x ) = ϕ ( x y ) = ϕ (0) = 0 by Proposition 3.2. This proves the proposition.
Remark. The converse of Proposition 3.3 need not be true in general. In Example 3.1, the mapping ϕ is an order reversing, but not an anti-homomorphism.
Let ( X , ∗, 0) be a d -algebra and let ϕ : X X be a map. We denote by 1 := ϕ (0).
Proposition 3.4. Let ( X , ∗, 0) be a d-algebra and let ϕ : X X be both order reversing and self-inverse . Then ( X , ∗, 0) is bounded.
Proof . Given x X , we have
• x∗ 1 =x∗ϕ(0) [1 =ϕ(0)]
• =ϕ(ϕ(x)) ∗ϕ(0) [ϕ: self-inverse]
• = 0 [ϕ: order reversing]
Let ( X , ∗, 0) be a d -algebra. We define a relation “≤” on X by x y if and only if x y = 0 for all x , y X . Note that the relation ≤ need not be a partial order on X . We define a relation “∧ on X by x y := x ∗ ( x y )) for all x , y X .
Proposition 3.5. Let ( X , ∗, 0) be a d-algebra . If ϕ : X X is self-inverse , then ϕ (1) = 0.
Proof . It follows from ϕ is self-inverse that 0 = ϕ ( ϕ (0)) = ϕ (1).
Theorem 3.6. Let ( X , ∗, 0) be a d-algebra and let ϕ : X X be a self-inverse map . If we define x y := ϕ [ ϕ ( y ) ∧ ϕ ( x )], then
ϕ ( x y ) = ϕ ( y ) ∨ ϕ ( x )
for all x , y X .
Proof . Given x , y X , we have
• ϕ(x∧y) =ϕ[ϕ(ϕ(x)) ∧ϕ(ϕ(y))] [ϕ: self-inverse]
• =ϕ[ϕ(a)) ∧ϕ(b)] [a=ϕ(x),b=ϕ(y)]
• =b∨a
• =ϕ(y) ∨ϕ(x)
Theorem 3.6 shows that the first De Morgan’s law implies the analog of the second De Morgan’s law and conversely, since x y y x in general. Moreover, it follows that x y = ϕ ( ϕ ( x y )) = ϕ [ ϕ ( y ) ∨ ϕ ( x )] for all x , y X .
Theorem 3.7. Let ( X , ∗, 0) be a d-algebra with PPT Slide
Lager Image
for all x X . If ϕ : X X is a self-inverse map , then x x = x and x x = x for all x X .
Proof . (i). Given x X , we have
• x∨x=ϕ[ϕ(x) ∧ϕ(x)] [Theorem 3.6]
• =ϕ[ϕ(x) ∗ (ϕ(x) ∗ϕ(x)]
• =ϕ(ϕ(x) ∗ 0) [(D1)]
• =ϕ(ϕ(x)) [(1)]
• =x[ϕ: self-inverse]
(ii). x x = x ∗ ( x x ) = x ∗ 0 = x .
Proposition 3.8. Let ( X , ∗, 0) be a d-algebra with PPT Slide
Lager Image
for all x , y , z X . Then x y x and x y y for all x , y X .
Proof . (i). Given x , y X , by applying (2), we obtain
• (x∧y) ∗a= (x∗ (x∗y)) ∗a
• = (x∗x) ∗ (x∗y)
• = 0 ∗ (x∗y)
• = 0
(ii). Given x , y X , we have ( x y )∗ y = ( x ∗( x y ))∗ y = ( x y )∗( x y ) = 0.
Theorem 3.9. Let ( X , ∗, 0) be a d-algebra with the condition ( 2 ). If ϕ : X X is a self-inverse anti-homomorphism , then x ∗ ( x y ) = 1 and y ∗ ( x y ) = 1 f or all x , y X .
Proof . (i). Since ϕ : X X is a self-inverse anti-homomorphism, for all x , y X , we obtain
• x∗ (x∨y) =x∗ϕ(ϕ(y) ∧ϕ(x))
• =x∗ϕ[ϕ(y) ∗ (ϕ(y) ∗ϕ(x))]
• =ϕ(ϕ(x)) ∗ϕ[ϕ(y) ∗ (ϕ(y) ∗ϕ(x))]
• =ϕ[[ϕ(y) ∗ (ϕ(y) ∗ϕ(x))] ∗ϕ(x)]
• =ϕ[[(ϕ(y) ∗ϕ(x)) ∗ (ϕ(y) ∗ϕ(x))]]
• =ϕ(0)
• = 1
and
• y∗ (x∨y) =ϕ(ϕ(x)) ∗ϕ[ϕ(y) ∗ (ϕ(y) ∗ϕ(x))]
• =ϕ[[ϕ(y) ∗ (ϕ(y) ∗ϕ(x))] ∗ϕ(y)]
• =ϕ[(ϕ(y) ∗ϕ(y)) ∗ (ϕ(y) ∗ϕ(x))]
• =ϕ(0)
• = 1
CONCLUSION
Whether such functions exists or not depends on the special properties of the d -algebras. BCK -algebras have the partial order structure, but d -algebras have no such a structure and so we need to seek another conditions for obtaining the analog of structures in d -algebras. This kind of functional approach can be connected with mirror d -algebras discussed in  in a new direction.
References